Is there a way to "rescue" mesh that has very bad topology?
To deal with bad topology, but not having to start from scratch, It makes sense creating a new mesh and retopologizing. That is: using some form of vertex/edge/face snapping and/or a shrinkwrap modifier, you trace and receate your old mesh using proper topology. This technique is also useful when sculpting at a high poly count and then re-creating the mesh with less polygons.

It sounds daunting but in reality it is not a big deal at all. 

Here are a few links that might help:


http://blog.digitaltutors.com/retopologize-nine-times-faster-shrink-wrap-modifier-blender/
https://www.blender.org/manual/modeling/meshes/editing/retopo.html
https://www.youtube.com/watch?v=eaqYqkjimD8